The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. These eight further families can be grouped in two classes of OP’s:
Hahn class (or linear lattice class). These are OP’s ${p}_{n}(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta}}_{x}$ or ${\nabla}_{x}$ or ${\delta}_{x}$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.
Wilson class (or quadratic lattice class). These are OP’s ${p}_{n}(x)={p}_{n}(\lambda (y))$ (${p}_{n}(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta}}_{y}}{{\mathrm{\Delta}}_{y}(\lambda (y))}$ or $\frac{{\nabla}_{y}}{{\nabla}_{y}(\lambda (y))}$ or $\frac{{\delta}_{y}}{{\delta}_{y}(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.
In addition to the limit relations in §18.7(iii) there are limit relations involving the further families in the Askey scheme, see §§18.21(ii) and 18.26(ii). The Askey scheme, depicted in Figure 18.21.1, gives a graphical representation of these limits.
The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek).
Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and standardization (§§18.2(i), 18.2(iii)) for the Hahn polynomials ${Q}_{n}(x;\alpha ,\beta ,N)$, Krawtchouk polynomials ${K}_{n}(x;p,N)$, Meixner polynomials ${M}_{n}(x;\beta ,c)$, and Charlier polynomials ${C}_{n}(x;a)$.
${p}_{n}(x)$ | $X$ | ${w}_{x}$ | ${h}_{n}$ | |
---|---|---|---|---|
Hahn | ${Q}_{n}(x;\alpha ,\beta ,N)$, $n=0,1,\mathrm{\dots},N$ | $\{0,1,\mathrm{\dots},N\}$ | $\frac{{\left(\alpha +1\right)}_{x}{\left(\beta +1\right)}_{N-x}}{x!(N-x)!}$, $\alpha ,\beta >-1$ or $$ | $\frac{{(-1)}^{n}{\left(n+\alpha +\beta +1\right)}_{N+1}{\left(\beta +1\right)}_{n}n!}{(2n+\alpha +\beta +1){\left(\alpha +1\right)}_{n}{\left(-N\right)}_{n}N!}$ If $$, then ${(-1)}^{N}{w}_{x}>0$ and ${(-1)}^{N}{h}_{n}>0$. |
Krawtchouk | ${K}_{n}(x;p,N)$, $n=0,1,\mathrm{\dots},N$ | $\{0,1,\mathrm{\dots},N\}$ | $\left({\displaystyle \genfrac{}{}{0pt}{}{N}{x}}\right){p}^{x}{(1-p)}^{N-x}$, $$ | ${\left({\displaystyle \frac{1-p}{p}}\right)}^{n}/\left({\displaystyle \genfrac{}{}{0pt}{}{N}{n}}\right)$ |
Meixner | ${M}_{n}(x;\beta ,c)$ | $\{0,1,2,\mathrm{\dots}\}$ | ${\left(\beta \right)}_{x}{c}^{x}/x!$, $\beta >0$, $$ | $\frac{{c}^{-n}n!}{{\left(\beta \right)}_{n}{(1-c)}^{\beta}}$ |
Charlier | ${C}_{n}(x;a)$ | $\{0,1,2,\mathrm{\dots}\}$ | ${a}^{x}/x!$, $a>0$ | ${a}^{-n}{\mathrm{e}}^{a}n!$ |
${p}_{n}(x)$ | ${k}_{n}$ |
---|---|
${Q}_{n}(x;\alpha ,\beta ,N)$ | $\frac{{\left(n+\alpha +\beta +1\right)}_{n}}{{\left(\alpha +1\right)}_{n}{\left(-N\right)}_{n}}$ |
${K}_{n}(x;p,N)$ | ${p}^{-n}/{\left(-N\right)}_{n}$ |
${M}_{n}(x;\beta ,c)$ | ${(1-{c}^{-1})}^{n}/{\left(\beta \right)}_{n}$ |
${C}_{n}(x;a)$ | ${(-a)}^{-n}$ |
These polynomials are orthogonal on $(-\mathrm{\infty},\mathrm{\infty})$, and with $\mathrm{\Re}a>0$, $\mathrm{\Re}b>0$ are defined as follows.
18.19.1 | $${p}_{n}(x)={p}_{n}(x;a,b,\overline{a},\overline{b}),$$ | ||
18.19.2 | $$w(z;a,b,\overline{a},\overline{b})=\mathrm{\Gamma}\left(a+iz\right)\mathrm{\Gamma}\left(b+iz\right)\mathrm{\Gamma}\left(\overline{a}-iz\right)\mathrm{\Gamma}\left(\overline{b}-iz\right),$$ | ||
18.19.3 | $$w(x)=w(x;a,b,\overline{a},\overline{b})={|\mathrm{\Gamma}\left(a+\mathrm{i}x\right)\mathrm{\Gamma}\left(b+\mathrm{i}x\right)|}^{2},$$ | ||
18.19.4 | $${h}_{n}=\frac{2\pi \mathrm{\Gamma}\left(n+a+\overline{a}\right)\mathrm{\Gamma}\left(n+b+\overline{b}\right){|\mathrm{\Gamma}\left(n+a+\overline{b}\right)|}^{2}}{\left(2n+2\mathrm{\Re}\left(a+b\right)-1\right)\mathrm{\Gamma}\left(n+2\mathrm{\Re}\left(a+b\right)-1\right)n!},$$ | ||
18.19.5 | $${k}_{n}=\frac{{\left(n+2\mathrm{\Re}\left(a+b\right)-1\right)}_{n}}{n!}.$$ | ||
These polynomials are orthogonal on $(-\mathrm{\infty},\mathrm{\infty})$, and are defined as follows.
18.19.6 | $${p}_{n}(x)={P}_{n}^{(\lambda )}(x;\varphi ),$$ | ||
18.19.7 | $${w}^{(\lambda )}(z;\varphi )=\mathrm{\Gamma}\left(\lambda +iz\right)\mathrm{\Gamma}\left(\lambda -iz\right){\mathrm{e}}^{(2\varphi -\pi )z},$$ | ||
18.19.8 | $$w(x)={w}^{(\lambda )}(x;\varphi )={\left|\mathrm{\Gamma}\left(\lambda +\mathrm{i}x\right)\right|}^{2}{\mathrm{e}}^{(2\varphi -\pi )x},$$ | ||
$\lambda >0$, $$, | |||
18.19.9 | ${h}_{n}$ | $={\displaystyle \frac{2\pi \mathrm{\Gamma}\left(n+2\lambda \right)}{{(2\mathrm{sin}\varphi )}^{2\lambda}n!}},$ | ||
${k}_{n}$ | $={\displaystyle \frac{{(2\mathrm{sin}\varphi )}^{n}}{n!}}.$ | |||
A special case of (18.19.8) is ${w}^{(1/2)}(x;\pi /2)=\frac{\pi}{\mathrm{cosh}\left(\pi x\right)}$.